Curriculum is more than pieces of information, more than subject matter, more even than the disciplines. Curriculum is an ongoing engagement with the problem of determining what knowledge and experiences are the most worthwhile. With each person and with each situation, that problem takes on different shadings and meanings. (Ayers 86, 1993)

 

Teaching and learning are not mutually exclusive. In order for us to be able to teach our student in the most successful and efficient manner, we as educators, must constantly be engaged in  learning about how to improve ourselves for the sake of our students and their success.

 

On all fronts we are trying to revamp and improve our curriculum so that student in the United States will be able to compete and compare to their peers around the world. (TIMSS 1995) Based on the Third International mathematics and Science Study (TIMSS) reports of 1999, US students were comparable to their peers up to fourth grade. However, once  in the middle school grades and high school, the US student rank very low in science and math compared to their peers from other industrialized nations.  As a result of this study many efforts have been and are being put forward in order to bring students here in the United States up to a level where they will be able to compete with their peers around the world.

 

In order to meet this goal the National Council of Teacher of Mathematics (NCTM) has published the Principle and Standards for School Mathematics (2000).  This document is based upon five main content strands: Problem Solving, Geometry, Algebra, Measurement, Data Analysis & Statistic. Each strand is taught at almost every grade level based upon what is developmentally appropriate for that particular group.

 

 In response to the NCTM Standards and Principals document, another organization, Achieve Inc.’s Mathematics Achievement Partnership (MAP), composed of member from both the private and public sector,  has published a document, entitled Foundations for Success Mathematics for the Middle Grades,  which also states specific content material that should be taught in Mathematics at the different grade levels.  The first draft focuses primarily on the middle school grades, sixth through eighth grades.  However, suggestions are given for what should be taught in the elementary grades and high school grades.

 

The purpose of this paper is to investigate the developmental appropriateness of the materials and suggestion for instruction in both of the above mentioned documents. Are the suggestion objectives and projects something that an average fifth grade would be able to perform successfully? How can we be sure that the average fifth grader would be interested in such activities?  Are the objectives and goals set in these documents realistic to achieve in the average fifth grade classroom?

 

 

What is developmentally appropriate for a fifth grader to learn?

 

Before we can take a closer look at the NCTM Principles and Standards and the Achieve documents, let us begin by understanding what is cognitively appropriate for an average nine or ten year old to learn.

 

Cognitive Theory of Development

 

Jean-Piaget’s Cognitive Development Theory states there are two basic tendencies that govern both physiological functions and mental functions : organization, which is the tendency to systematize and combine processes into coherent general systems, and adaptation, which is the tendency to adjust to the environment.  Intellectual process transform experiences into a form that the child can use in dealing with new situations, and in order to maintain a state of balance the intellectual processes seek a  balance through the process of equilibrium. Equilibrium is a from of self-regulation that all individuals use to bring coherence and stability to their conception of the world.

 

As children interact with their environment, parents, teachers, and peers they form organized patterns of behavior or thought called schemes. Schemes can be behavioral, for example, throwing a ball, or cognitive, e.g. realizing that there are many different kinds of balls.  Whenever a child encounters a new experience that does not fit into an existing scheme, adaptation is necessary.

 

Adaptation is the process of creating a good “fit” or match between one’s conception or reality and the real-life experiences one encounters; a tendency to adjust to the environment.   According to Piaget adaptation is accomplished by two subprocesses: assimilation and accommodation.  Assimilation  is when a new experience is fitted into an existing scheme, and accommodation is when an existing scheme is changed to or a new scheme is created to fit the new experience.  

 

An example of assimilation would be when a young child goes to the  aquarium for the first time and calls the minnows a “little fish” and the whales “big fish.” In both cases the child is assimilating – fitting a new experience into an existing scheme – in this case the conception that all animals that live in the water are fish.  When an adult points out that whales are mammals, not fish, the child begins to accommodate – to change her existing scheme to fit the new experience – then gradually a new scheme forms that contains nonfish animals that live in the water.

 

According to Piaget, in their desire in their desire to be organized, individuals try to have a place for  everything (accommodation) so that they can put everything in its place (assimilation). The product of organizing and adapting is the creation of new schemes that allow us to organize at a higher level and to adapt more effectively.

 

Piaget believes that individuals desire to organize their knowledge in order to achieve the best possible adaptation to their environment.  This is done through the process of equilibration, which is the tendency to organize schemes to allow better understanding of experiences.  However, in order for individuals to be driven to equilibration, there must exist a state of disequilibrium, or a perceived discrepancy between an existing scheme and something new. These processes are two sides of the learning coin: in order for equilibration to occur, disequilibrium must occur. Disequilibrium can occur spontaneously within an individual through maturation and experience, or someone else such as a teacher can stimulate it.

 

Therefore meaningful learning occurs when people create new ides, or knowledge from existing information .  (Biehler, Snowman,  1993)  In order for us to solve problems we have to use information from our memories that can be used to reach the solution.  Using information can mean experimenting, questions, reflecting, discovering, inventing and discussing.  This process of creating knowledge in order to solve a problem and eliminate a disequilibrium is called constructivism by Piagetian psychologists and educators.

 

In summary, the basic tenants of  Piaget’s Cognitive Theory of Development involves the following stages and processes:

·        Scheme are processed through organization and adaptation.

·        Adaptation involves the process of assimilation and accommodation in order to organize new information

·        Equilibration and disequilibration allows for better understanding of experiences.

 

Piaget states that unlike organization and adaptation, which are considered to be invariant functions due to the fat that these thought processes function the same way for infants, children, adolescents, and adults, schemes are not invariant.  Schemes undergo systematic change at particular points in time therefore there are real differences between the ways in which younger and older children think, and between the ways children and adults think. The schemes evolve through four primary stages: Sensorimotor, Prepoperational, Concrete operational, and Formal operational.

 

The Concrete Operational stage describes the characteristics of children within the seven to eleven years old range.  During this stage schemes are developed that allow for a greater understanding of such logic-based tasks as conservation ( matter is neither created nor destroyed but simply changes shape or form or position), class inclusion (constructing hierarchical relationships among related classes of items), and serration (arranging items in a particular order). However, operational thinking is limited to objects that are actually present or that children have experienced directly or concretely.  According Piaget, during this concrete operational stage a child between the ages of seven to eleven is capable of mentally reversing actions but generalizes only from concrete experiences. (Biehler, Snowman, 1993)

 

Cognitive Characteristics of Elementary Grades

 

Research has found that there exists six differences in cognitive functioning during the elementary school years.  Girls on average are superior in verbal fluency, spelling, reading, and mathematical computation. Boys, on average, are superior in mathematical reasoning, in tasks involving understanding of spatial relationships and in solving insight problems. (Biehler, Snowman, 1993) However more recent studies have found that the sex differences in cognitive abilities is getting smaller.

 

During the elementary grades there exists differences in cognitive styles amongst students.  Cognitive styles refers to tendencies or preferences to respond to a variety of intellectual tasks and problems in a particular fashion.  This is not the forum to go into detail about he different cognitive learning styles, however the different styles are reflective of the different types of learners that exist in any given classroom. Therefore the instruction and classroom environment should always try to meet the interests and needs of the different students.

 

Cognitive and Constructivist Approaches to Mathematics

 

Constructivist approaches to mathematics emphasize a deep understanding of concepts as opposed to memorization, discussion and explanation, and exploration of students’ implicit understandings.  Educators and psychologists who take the constructivists approach emphasize the importance of students’ construction of knowledge and minimal use of rote memorization. An important part of this approach is student discussion – asking questions and given explanation. 

 

Constructivist approaches to teaching recommend :complex, challenging learning environments; social negotiations and shared responsibility as part of learning; multiple representation of content; understanding that knowledge is constructed and student-centered instruction. (Woolfolk, 1998)

 

Jere Confry (1990) analyzed an expert mathematics teacher in a class and identified five components for a constructivist approach to Mathematics:

  1. Promote students’ autonomy and commitment to their answers.
  2. Develop students’ reflective processes
  3. Construct a case history of each student
  4. If the student is unable to solve a problem, intervene to negotiate a possible solution with the student.
  5. When the problem is solved, review the solution.    (Woolfolk   363,  1998)

 

NCTM Principle and Standards for School Mathematics

Goal of NCTM & Standard Based Curriculum

The National Council of Teachers of Mathematics (NCTM) is a nonprofit, nonpartisan education association that was founded in 1920.  The organization has more than 100 000 members and 250 Affiliates located throughout the United States and Canada. NCTM is dedicated to improving mathematics teaching and learning, kindergarten through high school, and facilitates ongoing dialogue and constructive discussion with educators about what is best for our students.

NCTM is committed to the view that standards can play a critical and leading role in guiding the improvement of mathematics education in this country.  The responsibility to ensure that all students receive a high quality mathematics education rests with the teachers of mathematics, school leaders, and parents.  All parties must work together in order to create a mathematics classroom where students of varied backgrounds and abilities work with expert teachers, learning important mathematical ideas with understanding, in environments that are “…equitable, challenging, supportive, and technologically equipped for the twenty-first century. (NCTM Principle and Standards  3, 2000)

Principles and Standards emphasizes the need for a common foundation of mathematics to be learned by all students.  This does not imply that all students are alike. Students exhibit different talents, abilities, achievements, needs and interests in mathematics.  Despite this all students must have equal access to the best quality mathematics instruction. Students with special educational needs must have the opportunities and support they require to attain a substantial understanding of important mathematics and students with a deep interest in mathematics and scientific careers must have their talents and interests engaged.  The goal must always focus on ultimately providing the student with the best possible mathematical experience and instruction.

Principle and Standards

In order to develop a well-rounded school mathematics program the NCTM document provides guidelines based on six Principles and ten Standards. The Principles describe particular features of high-quality mathematics education. The Standards describe the mathematical content and processes that students should learn. “Together, the Principles and Standards constitute a vision to guide educators as they strive for the continual improvement of mathematics education in classrooms, schools, and educational systems.” (NCTM Principle and Standards 5, 2000)

The six principles that describe overarching themes are:

These principles are very critical  to an effective, well-designed  school mathematics program. They can influence the development of curriculum frameworks, the selection of curriculum materials, the planning of instructional units or lessons, the design of assessments, the assignment of teachers and students to classes, instructional decisions in the classroom, and the establishment of supportive professional development programs for teachers. (NCTM Principle and Standards 17, 2000)

The Principle and Standards describes five content standards and five process standards that must appear in a mathematics program at each grade-band.  The five content standards are: Number & Operations, Algebra, Geometry, Measurement and Data Analysis & Probability. The five process standards are: Problem Solving, Reasoning & Proof, Communication, Connections, and Representations.

Mathematical Content

The Standards presents a view of mathematics learning, teaching, and assessment that shifts the focus of curriculum and instruction. Unlike the traditional mathematics education that focused on memorization, rote learning , and the application of facts and procedures, “..the Standards-based approach emphasizes the development of conceptual understanding and reasoning.” (Goldsmith, June, 41, 1998)

There has been a pedagogical shift which has moved the focus from direct instruction, drill and practice toward more active student engagement with mathematical ideas through  collaborative  learning, hands-on explorations, the use of multiples representations and discussion and writing. This view of having the students build their own knowledge is referred to as “constructivist”. As was stated previous, Piaget’s cognitive learning theory stress the constructivist approach to learning because it promotes deeper and more substantial understanding.

The standards stress the importance of helping students develop deep conceptual understanding relating to the major strands of mathematics which as stated previously are: number and operation; patterns, functions and algebra; geometry and measurement; and data analysis, statistics, and probability.  In addition to the promoter a deeper conceptual understanding, the standards also stress that students must acquire fluency with skill-based manipulations, and learn to reason and communicate about mathematical ideas.

Math is not presented as a set of discrete and unrelated topics that students learn , forget after the test and perhaps relearn the next year.  These curriculum support students’ development of mathematical understanding by requiring them to hypothesize, predict, observe, and reason about mathematical situations. 

Mathematical Processes

According to the NCTM document Principle and Standards, students gain mathematical competence by learning to work with mathematical ideas, to solve, problems an to communicate their ideas to others. The standards promote that curriculum programs should develop the following five mathematical processes:

Problem Solving – students use mathematically productive ways to approach problems, which includes hypothesizing, building a variety of representations, abstracting, and making generalizations.

Reasoning and Proof – students think systematically  and critically about mathematics by making observations, proposing and investigating conjectures, and developing mathematical arguments and proof.

Communication – students effectively organize and articulate their thinking, consider the ideas of their peers and others, and develop use and fluency with the language of mathematics.

Connections – students recognize the coherence of mathematics as a discipline by seeing interrelations among ideas and by understanding the power of mathematics through connections with outside disciplines and contexts. (real-world connections)

Representations – students develop and should effectively use a repertory of representations to organize thinking and to model and interpret mathematical situations.

Through emphasizes on these processes, the Standards stress that mathematical thinking develops through engagement, inquiry and exploration in mathematical work. The stress and focus on engaging student in doing mathematics is intended to help student understand the why as well as the how of the mathematics they study.  In order to support the students’ construction of deep and flexible understanding to math content, it is recommended that student across the grades:

Interact with a range of materials for representing problem situations, such as manipulatives, calculators, computers, diagrams, tables, and charts;

            Work collaboratively as well as individually;

Discuss mathematical ideas; and

Focus on making sense of the mathematics they are studying as well as on learning to achieve accurate and efficient solutions to problems.   (Goldsmith, Mark, 1999)

The focus of the standards is to facilitate the development of programs the support teachers in creating classrooms where such work as listed above can occur by offering lessons and activities that motivate a wide range of student an engage them in the study of powerful mathematical ideas.  A standard based program shares some of the following characteristics:

§         Standards-based materials take an integrated approach to topics from the earliest grades, with several areas of mathematics appearing at each grade level and developing connections to one another.  Skill acquisition and practice is embedded in activities other than pure drill.

§         Mathematical ideas reappear at different grade levels in increasingly sophisticated forms. For example, exploring patterns at the elementary levels builds the foundation for the study of algebraic relationships in the upper grades.

§         Mathematical knowledge is developed within both practical and conceptual contexts, with less emphasis on. “... rote symbol or number manipulation.”

§         Problems presented are complex, involving a number of mathematical ideas and skills and requiring more time and thought to solve than previous problems of the past.

§         The programs emphasize different kinds of representations such as chards, tables, graphs, diagrams, and formal notation for exploring, describing and testing problem situations.

§         Lessons use less direct instruction and more student collaboration, conjecture, exploration, and discussion of mathematical ideas and these lessons can extend over several days and involve student activity followed by class discussion.

Developmental Appropriateness

In addition to these features of a standards based curriculum, one of the goals stated in the introduction of the Principle and Standards document is  “School mathematics programs should not address every topic every year. Instead, students will reach certain levels of conceptual understanding and procedural fluency by certain points in the curriculum.” (NCTM Principle and Standards 6, 2000)  Based on this statement it is clear that the NCTM Principle and Standards document acknowledges the fact that students will understand specific material at specific points in their development. 

Comparing the suggestions, activities, goals and objectives of the standards to what is known to be developmentally appropriate for an average fifth grader, the standards align themselves in a very organized manner so that they do fulfill the cognitive needs for not only that age group, but all age groups from Kindergarten through twelfth grade.

Let us take a closer look at a suggested activity in the Geometry standard for the third to fifth grade band.   In this strand one of the instructional outcomes from prekindergarten through grade twelve is to enable all students to:

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Within this objective, all students in grades three through five should be able to

•

identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes;

•

classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids;

•

investigate, describe, and reason about the results of subdividing, combining, and transforming shapes;

•

explore congruence and similarity;

•

make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions

Keeping in mind that the activities suggesting within the strands for each grade band are building upon knowledge and information that the students have gained in the lower grades.  In the lower grades students have already sorted and classified general objects such as cylinders by identifying general characteristics. The following text from the NCTM Standards document presents an example for this objective that appropriately fits the cognitive stage of Concrete operational for this age group.

In grades 3–5, they should develop more-precise ways to describe shapes, focusing on identifying and describing the shape's properties and learning specialized vocabulary associated with these shapes and properties. To consolidate their ideas, students should draw and construct shapes, compare and discuss their attributes, classify them, and develop and consider definitions on the basis of a shape's properties, such as that a rectangle has four straight sides and four square corners. For example, many students in these grades will easily name the first two shapes in figure 5.10 as rectangles but will need to spend more time discussing why the third one is also a rectangle—indeed, a special kind of rectangle.  (NCTM Principle and Standards 164, 2000)

Fig. 5.10. Examples of rectangle

Through explorations, discussions and formulation of mathematical arguments, the students will formulate conjectures about geometric properties and relationships.  Using other tools such as geometry software, concrete materials, drawings, they can develop and test their ideas and discover why some geometric definitions are true.

 

This example is a clear representation that the NCTM Principle and Standards document is focused on promoting the age appropriate cognitive learning ideals and goals.

 

The Problem Solving Standard for grades three through five also represents the constructivist approach to learning and problem solving.  In grades three through five students should have frequent experiences with problems that interest, challenge, and engage them in thinking about important mathematics.  “Problem solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned” (NCTM Principle and Standards   182, 2000)

 

Through the challenge and experiencing new situations disequilibrium will motivate student to assimilate and accommodate the new information so that they can reach a state of equilibrium again.

 

The following example is a sample of the type of problems that can be used to promote problem solving skills and processes throughout the curriculum.

If you roll two number cubes (both with the numbers 1–6 on their faces) and subtract the smaller number from the larger or subtract one number from the other if they are the same, what are the possible outcomes? If you did this twenty times and created a chart and line plot of the results, what do you think the line plot would look like? Is one particular difference more likely than any other differences? (NCTM Principle and Standards  183, 2000)

This problem was tried in a classroom and the solution to this problem was  not immediately obvious. The students have to generate and organize information and then evaluate and explain the results. The teacher was able to introduce notions of probability such as predicting and describing the likelihood of an event, and the problem was accessible and engaging for every student. It also provided a context for encouraging students to formulate a new set of questions.

Good problems and problem-solving tasks encourage reflection and communication and can emerge from the students' environment or from purely mathematical contexts. They generally serve multiple purposes, such as challenging students to develop and apply strategies, introducing them to new concepts, and providing a context for using skills. Good problems are also the foundation for encouraging and promoting meaningful learning so that students will be involved in constructing their own knowledge and understanding.

The problem solving process involves the ability to activate relevant schemes (organized collection of facts, concepts, principles and procedures) from long-term memory when they are needed. Learning to solve problems means learning to find problems, represent them, draw on prior knowledge, formulate solutions, and evaluate solutions. (Biehler, Snowman 1993)  This is a central aspect of the constructivist approach to learning and teaching. 

Students within this grade level have developed multiple cognitive learning styles, and the student centered approach to problem-solving, learning and teaching, allows for the different styles to benefit and learn from each other. (Biehler, Snowman, 1993)

The NCTM Principle and Standards are based upon and reflect what is cognitively appropriate for each grade-level.

Foundations for Success Mathematics for the Middle Grades

In order to confront the challenges of improving the mathematics standards across the county, Achieve Inc.’s Mathematics Achievement Partnership (MAP) states that improving student performance depends on a “comprehensive approach based on:                

§         Supporting teachers by equipping them with the knowledge and skills they need to help raise student proficiency;

§         Measuring student proficiency on a regular basis

§         Using assessment result to assist teachers and improve classroom practice. (Achieve Inc. 2001)

MAP’s work is based on the results frorn the 1995 Third International Mathematics and Science Study.  The data from this and the 1999 follow-up, TIMSS-R showed that the U.S. students performed significantly below their peers around the world. The U.S. students were amongst the highest ranked in mathematics up to the elementary grades. The scores begin to fall in middle school and continue to fall at the high school level. Based upon the TIMSS data, the mathematics curricula in grades six through eight do not provide a deep study in the previously taught concepts form the elementary grades.

The result of the TIMSS study and Achieve’s own analysis of 21 state tests of fourth and eighth grade students, confirmed that strong performance in mathematics would require detailed standards that emphasize procedural skills, conceptual knowledge and problem solving.  The NCTM Principle and Standards set the stage for the Achieve analysis and document of Foundations for Success.

Foundations for Success is a blueprint for mathematics in the middle grades that is benchmarked to international standards. The goal is to provide student with a strong foundation  in mathematics in middle school so that they will have the necessary tools to succeed in high school, college and work place.

This document does not go into detailed sample problems and possible student solutions. Most of the problems and their solutions that are presented are there for the purpose of providing a curriculum guide of the topics to be covered within each math content at each grade level and “…to illustrate the scope, depth, and meaning of the expectations” (Foundations for Success, 2001)  The current draft calls for student to cover more mathematics than most U.S. students are learning right now by the end of eighth grade or even by the end of high school.

However ,MAP’s expectations are based upon the fact that every child is provided with adequate support, included strong preparation from grades K through five.

Elementary Grades

In appendix B of the Foundations for Success document the following are some of the topics and MAP expectations  stated for grades one through five:

            Number:

§         Understand the relationship between numbers, quantities and place value in whole numbers

§         Understand and fluently perform the basic arithmetic computations with integers, decimals and fractions

§         Understand fractions and decimals are two different representations of the same concept, and be able to convert among equivalent forms of the same number.

§         Understand the concept of the number line and the location on it of integers, fractions, mixed numbers, and decimals, both positive and negative.

Algebra and Functions

§         Develop number sentences for problem situations

§         Recognize and use the commutative, distributive and associative properties

§         Recognize graphs expressing functional relations

Measurement and Geometry:

§         Use common and nonstandard units to measure objects

§         Select appropriate units for a given measurement task

§         Identify and classify common geometric figures

§         Understand from physical models simple volume and area relationships among geometric figures

Data Analysis:

§         Measure and count a wide variety of physical  objects

§         Organize and interpret numerical and categorical data

§         Construct simple graphs and charts from tables of data

§         Calculate and understand the meaning of mean, median, mode and range of numerical data.

Just looking at this list of expectations for grades one through five, it is difficult to make any kind of assessment as to the developmental appropriateness of the material for fifth grade.  However, looking at this document as it discusses the standards for middle school and briefly comparing it to the NCTM Principle and Standards document, the MAPs’s Foundation for Success does not go into as much detail of the pedagogy, learning and teaching styles.  At the beginning of each content area there is a summary of the teacher and student expectations and a brief description of the content. The rest of the section has sample problems and solutions that are not student samples.

Recommendations

The NCTM Principle and Standards document presented a very organized and systematic explanation of what is to be expected from the instructors and learners regarding the principles and standards.  Each content and process strand has student examples that provide a clear guide for the curriculum developer, teacher and general educator.

Unlike the NCTM document, the MAP’s Foundation for Success document is not as detailed in stating their goals and expectations. Part of the problem arises from the fact the only copy in print currently is a draft version.  In addition to this, the draft focuses on the standards for middle school grades six through eight.

We have to wait and see what the MAP’s document for elementary school will look like, however they many consider developing a website or CD the has some sample problems and projects that can be done to meet the requirements of the content standards at the different grade levels.  In addition to this the document should include some discussion about the philosophy of what type of teaching and learning method is being promoted. Is it direct instruction? Constructivism? Inquiry based?

We shall have to wait to see what is published before we can make a far assessment of the developmental appropriateness of the objectives suggested for fifth grade.

Which document is developmentally appropriate?

Until a more detail version of the Foundations for Success for elementary school is printed, it would only be fair to say that the NCTM Principle and Standards document is developmentally appropriate for fifth grade. Due to the emphasizes on the constructivist approach to teaching and learning, every student at every grade level can hope to use their own background knowledge and understanding to further their own learning. The student who needs to be challenge can be challenged and the student who needs extra assistance can get the extra assistance and guidance. 

As for the MAP document, it is difficult to make any kind of comparison until more detail is provided.

Despite whether the NCTM document or the MAP document is developmentally appropriate for fifth grade, we cannot ignore that fact that in order to be able to improve the standards and quality of our mathematics education we must try to provide all students in public school, regardless of their socio-economic and ethnic background with proper qualified instruction with appropriate resources.

Without the proper and qualified instruction and the resources it is difficult to provide our students with the intellectually stimulating environment in which they can explore, engage and construct their knowledge and understanding.

The Standards are a beginning. We must follow up the ideas and goals with action. Curriculum support materials must be designed to meet the standards.  Teachers must be instructed with the appropriate content and pedagogical knowledge. Resources must be made readily available to teachers and students.  Then we can engage ourselves in the real battle of teaching, learning, and assessing.

 

 

 

 

 

 

 

 

Developmental Appropriateness of the NCTM Principle and Standards vs. MAP’s Foundations of Success

 

 

 

 

 

 

 

 

 

 

 

 

 

Aisha Elahi

EDCI 650

December 16th, 2001

 

References

Achieve Inc. Mathematics Achievement Partnership. (2001)  Foundations for success mathematics for the middle grades.

Ayers, W. , (1993). To teach the journey of a teacher .  New York: Teachers College Press.

Biehler, R.F. , Snowman, J. , Psychology applied to teaching .  Dallas: Houghton Mifflin Company.

Copes, L. (2000) . Messy monk mathematics: an NCTM standards-inspired class. Mathematics Teacher, 93, 292-298.

Goldsmith, L., Mark, J. (1999) What is a standards-based mathematics curriculum? Educational Leadership, 57, 40-44.

National Council of Teachers of Mathematics. (2000) . Standard for school mathematics. [On-line]  Principles and standards for school mathematics.

Romber, T. (1999) . Comments: NCTM’s curriculum and evaluation standards. Teachers College Record, 100, 19-20.

Woolfolk, A.E. , (1998) . Educational psychology . Boston: Allyn and Bacon.

Third international mathematics and science study-repeat. (n.d.) http://nces.ed.gov/TIMSS/timss-r/index.asp